# tgamma, tgammaf, tgammal

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< c‎ | numeric‎ | math

Defined in header ``<math.h>`` |  |
float tgammaf( float arg ); | (1) | (since C99)
double tgamma( double arg ); | (2) | (since C99)
long double tgammal( long double arg ); | (3) | (since C99)
Defined in header ``<tgmath.h>`` |  |
#define tgamma( arg ) | (4) | (since C99)

1-3) Computes the gamma function of arg.
4) Type-generic macro: If arg has type long double, `tgammal` is called. Otherwise, if arg has integer type or the type double, `tgamma` is called. Otherwise, `tgammaf` is called.

### Parameters

arg | - | floating-point value

### Return value

If no errors occur, the value of the gamma function of arg, that is \(\Gamma(\mathtt{arg}) = \displaystyle\int_0^\infty\!\! t^{\mathtt{arg}-1} e^{-t}\, dt\)∫∞
0targ-1
e-t dt, is returned.

If a domain error occurs, an implementation-defined value (NaN where supported) is returned.

If a pole error occurs, ±HUGE_VAL, `±HUGE_VALF`, or `±HUGE_VALL` is returned.

If a range error due to overflow occurs, ±HUGE_VAL, `±HUGE_VALF`, or `±HUGE_VALL` is returned.

If a range error due to underflow occurs, the correct value (after rounding) is returned.

### Error handling

Errors are reported as specified in `math_errhandling`.

If arg is zero or is an integer less than zero, a pole error or a domain error may occur.

If the implementation supports IEEE floating-point arithmetic (IEC 60559):

- If the argument is ±0, ±∞ is returned and FE_DIVBYZERO is raised.
- If the argument is a negative integer, NaN is returned and FE_INVALID is raised.
- If the argument is -∞, NaN is returned and FE_INVALID is raised.
- If the argument is +∞, +∞ is returned.
- If the argument is NaN, NaN is returned.

### Notes

If arg is a natural number, tgamma(arg) is the factorial of arg - 1. Many implementations calculate the exact integer-domain factorial if the argument is a sufficiently small integer.

For IEEE-compatible type double, overflow happens if 0 < x < 1/DBL_MAX or if x > 171.7.

POSIX requires that a pole error occurs if the argument is zero, but a domain error occurs when the argument is a negative integer. It also specifies that in future, domain errors may be replaced by pole errors for negative integer arguments (in which case the return value in those cases would change from NaN to ±∞).

There is a non-standard function named `gamma` in various implementations, but its definition is inconsistent. For example, glibc and 4.2BSD version of `gamma` executes `lgamma`, but 4.4BSD version of `gamma` executes `tgamma`.

### Example

Run this code

    #include <errno.h>
    #include <fenv.h>
    #include <float.h>
    #include <math.h>
    #include <stdio.h>
    // #pragma STDC FENV_ACCESS ON

    int main(void)
    {
        printf("tgamma(10) = %f, 9!=%f\n", tgamma(10), 2 * 3 * 4 * 5 * 6 * 7 * 8 * 9.0);
        printf("tgamma(0.5) = %f, sqrt(pi) = %f\n", tgamma(0.5), sqrt(acos(-1)));

        // special values
        printf("tgamma(+Inf) = %f\n", tgamma(INFINITY));

        // error handling
        errno = 0; feclearexcept(FE_ALL_EXCEPT);
        printf("tgamma(-1) = %f\n", tgamma(-1));
        if (errno == ERANGE)
            perror("    errno == ERANGE");
        else
            if (errno == EDOM)   perror("    errno == EDOM");
        if (fetestexcept(FE_DIVBYZERO))
            puts("    FE_DIVBYZERO raised");
        else if (fetestexcept(FE_INVALID))
            puts("    FE_INVALID raised");
    }

Possible output:

    tgamma(10) = 362880.000000, 9!=362880.000000
    tgamma(0.5) = 1.772454, sqrt(pi) = 1.772454
    tgamma(+Inf) = inf
    tgamma(-1) = nan
        errno == EDOM: Numerical argument out of domain
        FE_INVALID raised

### References

- C23 standard (ISO/IEC 9899:2024):

  - 7.12.8.4 The tgamma functions (p: 250)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - F.10.5.4 The tgamma functions (p: 525)

- C17 standard (ISO/IEC 9899:2018):

  - 7.12.8.4 The tgamma functions (p: 250)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - F.10.5.4 The tgamma functions (p: 525)

- C11 standard (ISO/IEC 9899:2011):

  - 7.12.8.4 The tgamma functions (p: 250)

  - 7.25 Type-generic math <tgmath.h> (p: 373-375)

  - F.10.5.4 The tgamma functions (p: 525)

- C99 standard (ISO/IEC 9899:1999):

  - 7.12.8.4 The tgamma functions (p: 231)

  - 7.22 Type-generic math <tgmath.h> (p: 335-337)

  - F.9.5.4 The tgamma functions (p: 462)

### See also

lgammalgammaflgammal (C99)(C99)(C99) | computes natural (base-e) logarithm of the gamma function (function)
C++ documentation for tgamma

### External links

Weisstein, Eric W. "Gamma Function." From MathWorld — A Wolfram Web Resource.

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